Skip to main content
SpringerLink
Log in

A Comprehensive Introduction to the Theory of Word-Representable Graphs

  • Conference paper
  • First Online:
Developments in Language Theory (DLT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10396))

Included in the following conference series:

Abstract

Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word \(xyxy\cdots \) (of even or odd length) or a word \(yxyx\cdots \) (of even or odd length). A graph \(G=(V,E)\) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if \(xy\in E\).

Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-represent-able graphs including the most recent developments in the area.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The patterns considered in this section are ordered, and their study comes from Algebraic Combinatorics. There are a few results on word-representable graphs and (unordered) patterns studied in Combinatorics on Words, namely on squares and cubes in words, that are not presented in this paper, but can be found in [17, Sect. 7.1.3]. One of the results says that for any word-representable graph, there exists a cube-free word representing it.

References

  1. Akrobotu, P., Kitaev, S., Masárová, Z.: On word-representability of polyomino triangulations. Siberian Adv. Math. 25(1), 1–10 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, T.Z.Q., Kitaev, S., Sun, B.Y.: Word-representability of face subdivisions of triangular grid graphs. Graphs Comb. 32(5), 1749–1761 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, T.Z.Q., Kitaev, S., Sun, B.Y.: Word-representability of triangulations of grid-covered cylinder graphs. Discr. Appl. Math. 213(C), 60–70 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Collins, A., Kitaev, S., Lozin, V.: New results on word-representable graphs. Discr. Appl. Math. 216, 136–141 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, A., Kitaev, S., Zhang, P.: On 132-representable graphs. arXiv:1602.08965 (2016)

  6. Glen, M.: Colourability and word-representability of near-triangulations. arXiv:1605.01688 (2016)

  7. Glen, M.: Software. http://personal.cis.strath.ac.uk/sergey.kitaev/word-representable-graphs.html

  8. Glen, M., Kitaev, S.: Word-representability of triangulations of rectangular polyomino with a single domino tile. J. Comb. Math. Comb. Comput. (to appear)

    Google Scholar 

  9. Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Graphs capturing alternations in words. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 436–437. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14455-4_41

    Chapter  Google Scholar 

  10. Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Alternation graphs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 191–202. Springer, Heidelberg (2011). doi:10.1007/978-3-642-25870-1_18

    Chapter  Google Scholar 

  11. Halldórsson, M., Kitaev, S., Pyatkin, A.: Semi-transitive orientations and word-representable graphs. Discrete Appl. Math. 201, 164–171 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jones, M., Kitaev, S., Pyatkin, A., Remmel, J.: Representing graphs via pattern avoiding words. Electron. J. Comb. 22(2), 2.53 (2015). 20 pp

    MathSciNet  MATH  Google Scholar 

  13. Kim, J., Kim, M.: Graph orientations on word-representable graphs (in preparation)

    Google Scholar 

  14. Kitaev, S.: Patterns in Permutations and Words. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  15. Kitaev, S.: On graphs with representation number 3. J. Autom. Lang. Comb. 18(2), 97–112 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Kitaev, S.: Existence of \(u\)-representation of graphs. J. Graph Theor. 85(3), 661–668 (2017)

    Article  Google Scholar 

  17. Kitaev, S., Lozin, V.: Words and Graphs. Springer, Heidelberg (2015)

    Book  MATH  Google Scholar 

  18. Kitaev, S., Pyatkin, A.: On representable graphs. J. Autom. Lang. Comb. 13(1), 45–54 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Kitaev, S., Salimov, P., Severs, C., Úlfarsson, H.: On the representability of line graphs. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 478–479. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22321-1_46

    Chapter  Google Scholar 

  20. Kitaev, S., Seif, S.: Word problem of the Perkins semigroup via directed acyclic graphs. Order 25(3), 177–194 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mandelshtam, Y.: On graphs representable by pattern-avoiding words. arXiv:1608.07614 (2016)

  22. Pretzel, O.: On graphs that can be oriented as diagrams of ordered sets. Order 2(1), 25–40 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thomassen, C.: A short list color proof of Grötzsch’s theorem. J. Comb. Theor. Ser. B 88(1), 189–192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, UK

    Sergey Kitaev

Authors
  1. Sergey Kitaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Kitaev .

Editor information

Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Émilie Charlier

  2. Université de Liège, Liège, Belgium

    Julien Leroy

  3. Université de Liège, Liège, Belgium

    Michel Rigo

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Kitaev, S. (2017). A Comprehensive Introduction to the Theory of Word-Representable Graphs. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62809-7_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62808-0

  • Online ISBN: 978-3-319-62809-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation